An Insight on the Microwave Circulator Theory

This contribution is devoted to clarifying the analytical derivation of the microwave circulator matrixes. Although this topic seems well-acknowledged, the approach of many books is affected by an initial error that, in turn, affects the final result. It is worth noting that the error is not typographical, but a direct consequence of an incorrect assumption reported in many books. Starting from the knowledge that a three-port network cannot be simultaneously lossless, matched and reciprocal, the solutions of the system are usually calculated by verifying the unitary conditions on a matched matrix, which is the point where the error arises in many books. As a matter of fact, it should not be underestimated because microwave and millimeter-wave circulators are unique and basic components used in a huge number of applications due to their great ability to redirect microwave signals, and the related theory is covered within most of the microwave engineering courses. In view of that, it is of utmost importance clarifying the related aspects by providing all the mathematical steps to obtain the final matrixes of circulator.